Fractions, decimals and the number line

Fractions: what the operations do

A fraction $\frac{a}{b}$ is $a$ divided by $b$. The four operations follow from that.

• Add or subtract over a common denominator: $\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$.
• Multiply straight across: $\frac{2}{3} \cdot \frac{9}{10} = \frac{18}{30} = \frac{3}{5}$. Cancel common factors first to keep numbers small.
• Divide by multiplying by the reciprocal: $\frac{3}{4} \div \frac{3}{8} = \frac{3}{4} \cdot \frac{8}{3} = 2$. Dividing by a number between 0 and 1 makes the result larger.

Comparing fractions

Three methods, from quickest to most general:

1. Benchmarks. Compare each fraction with $\frac{1}{2}$ or 1. $\frac{5}{11}$ is below $\frac{1}{2}$ and $\frac{4}{7}$ is above, so $\frac{4}{7}$ is larger.
2. Cross-multiplication. For positive fractions, $\frac{a}{b} > \frac{c}{d}$ exactly when $ad > bc$. To compare $\frac{5}{8}$ with $\frac{7}{11}$: $5 \cdot 11 = 55$ and $8 \cdot 7 = 56$, so $\frac{5}{8} < \frac{7}{11}$.
3. Decimals. Divide out to two or three places; the on-screen calculator can do this in Quantitative Reasoning, but cross-multiplication is often faster.

A larger numerator alone, or a smaller denominator alone, tells you nothing. $\frac{5}{8}$ has the larger denominator of $\frac{5}{8}$ and $\frac{3}{5}$, yet it is the larger fraction.

Terminating and repeating decimals

Every fraction gives a decimal that either stops or repeats. Which one depends only on the denominator after the fraction is reduced:

• if the reduced denominator has no prime factors other than 2 and 5, the decimal terminates;
• otherwise it repeats forever.

So $\frac{7}{40}$ terminates ($40 = 2^3 \cdot 5$; it is 0.175), and $\frac{9}{24}$ terminates too, because it reduces to $\frac{3}{8} = 0.375$. But $\frac{7}{12}$ repeats ($12 = 2^2 \cdot 3$; it is $0.58333\ldots$).

The reason: a terminating decimal is a fraction over a power of 10, and $10 = 2 \cdot 5$, so only 2s and 5s can be made into a power of 10.

To find a far-off digit of a repeating decimal, find the length of the repeating block, divide the position by that length, and use the remainder. If $\frac{3}{7} = 0.\overline{428571}$ (block of 6), the 100th digit is at remainder $100 - 96 = 4$ within the block, which is 5. A remainder of 0 means the last digit of the block.

Place value

In 3,507.284 the 2 is in the tenths place, the 8 in the hundredths and the 4 in the thousandths. Rounding to a place looks only at the digit just to its right: 3,507.284 rounded to the nearest hundredth is 3,507.28. Multiplying by 10 moves every digit one place left, which is why $0.0375 \times 1000 = 37.5$.

The number line

Numbers increase to the right. On the negative side this reverses what feels natural about size: $-7 < -2$, because $-7$ is further left, even though 7 is larger than 2.

The distance between $a$ and $b$ is $|a - b|$. From $-2.4$ to $3.6$ it is $|3.6 - (-2.4)| = 6$. This is also what absolute value means: $|x|$ is the distance from $x$ to 0, so $|x| < 3$ describes the points within 3 of zero, the interval $-3 < x < 3$.

The midpoint of $a$ and $b$ is their average, $\frac{a + b}{2}$. More generally, the point a fraction $t$ of the way from $a$ to $b$ is

$$a + t(b - a)$$

Start at $a$, then move a fraction $t$ of the full distance towards $b$. If $b$ is to the left of $a$, then $b - a$ is negative and the formula moves left automatically.

Reciprocals and order

For two positive numbers, taking reciprocals reverses the order: $2 < 5$ but $\frac{1}{2} > \frac{1}{5}$. The same happens for two negatives. For one positive and one negative number, the order is kept, because a reciprocal never changes sign. Number-line questions that place $x$, $\frac{1}{x}$, $x^2$ and $-x$ test this along with the size facts for numbers between 0 and 1.

A habit that saves marks

Before comparing, write every value in one form, all fractions or all decimals, and mark each one on a sketch of the number line. Errors with negatives almost always come from comparing sizes without signs.